Suppose $(M,g)$ is a connected closed Riemannian manifold and $f:M\to [0,1]$ a smooth function. Suppose the critical points of $f$ consists of exactly points of $f^{-1}(0)$ (minimum) and $f^{-1}(1)$ (maximum). That is, $p\in M$ is a critical point of $f$ if and only if $f(p)$ is either $0$ or $1$. Also suppose that $f^{-1}(0)$ and $f^{-1}(1)$ are embedded submanifolds. Then is it true that the complement $M-f^{-1}(0)$ deformation retracts onto $f^{-1}(1)$?
Maybe we should consider the gradient flow $\varphi_t$ of $f$. (Let us assume that each integral curve converges as $t\to \infty$.) Since $f$ increases along the flow of $f$, every point in $M-f^{-1}(0)$ would approach to a point of $f^{-1}(1)$. But the flow never reaches a point of $f^{-1}(1)$, so the flow don't give directly a deformation retraction to $f^{-1}(1)$. Can we choose a tubular neighborhood of $f^{-1}$ of the form $f^{-1}[1-\epsilon,1]$?
